**26**

votes

**2**answers

3k views

### Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
$$\Lambda(n)=...

**3**

votes

**0**answers

52 views

### Convolution of decaying polynomials [closed]

I conjecture that if the functions $f$, $g$ defined on $\mathbb{R}^n$ satisfying
$$|f(x)| ≤ A(1+|x|)^{−M}, \quad |g(x)| ≤ B(1+|x|)^{−N}$$for some
$M$, $N > n$, then$$|(f * g)(x)| ≤ ABC(1+|x|)^{−L},$...

**7**

votes

**1**answer

379 views

### Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved).
Then the Fourier transform of this function is given by
$...

**1**

vote

**0**answers

24 views

### Extension of $\ell^{2}$ decoupling to $C^{2}$ hypersurfaces

Let $\phi : [0,1]^{n-1} \rightarrow \mathbb{R}$ be a $C^{2}$ function such that $\phi(0)=\nabla\phi(0)=0$ and $0<\frac{1}{C}<\nabla^{2}\phi<C$. Consider the compact $C^{2}$ hypersurface $S \...

**8**

votes

**2**answers

696 views

### Error of Discrete Fourier Transform on Finite Domain (vs. Continuous FT) in terms of Sobolev order

My question is about quantifying the error that occurs by approximating the continuous Fourier transform on a finite domain by using a discretised version with resolution $N$ for a function of a given ...

**25**

votes

**6**answers

4k views

### Explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros.
Because there are various explicit formulae ...

**5**

votes

**1**answer

296 views

### Application of Factorization Theory to Oscillatory Integral Estimates

In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form
$$S_{N}g(x):=\int_{\mathbb{R}^{n}}g(y)e^...

**21**

votes

**2**answers

7k views

### Truth of the Poisson summation formula

The Poisson summation says, roughly, that summing a smooth $L^1$-function of a real variable at integral points is the same as summing its Fourier transform at integral points(after suitable ...

**1**

vote

**0**answers

50 views

### A Global Restriction Estimate from Local Estimate

Let $S$ be a smooth hypersurface in $\mathbb{R}^{n}$ with surface measure $d\sigma$. Let $1\leq p,q\leq\infty$, $R>0$, and $\mathcal{N}_{R^{-1}}(S)$ denote the $R^{-1}$ neighborhood of $S$. Suppose ...

**6**

votes

**1**answer

129 views

### Decay of Fourier coefficients for compact Lie groups

Let $G$ be a compact Lie group, $G^\natural$ the space of conjugacy classes in $G$ with the natural pushforward of $G$'s Haar measure. Let $f\in L^2(G^\natural)$. Then the Peter–Weyl Theorem tells us ...

**32**

votes

**6**answers

6k views

### Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...

**5**

votes

**1**answer

68 views

### Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that
$$
\lvert u\rvert_s = \...

**1**

vote

**1**answer

153 views

### trigonometric series with nonnegative coefficients and prescribed values

Given real numbers $x_1$, $\dots$, $x_n$ and $r_1$, $\dots$, $r_n$ (with reasonable restrictions), is there a trigonometric series $T(x)=\sum_k a_k \cos(kx)$ with $a_k\ge 0$ such that
$$
|T(x_i)-r_i|&...

**2**

votes

**0**answers

100 views

### Decoupling Implies Local Decoupling

My question comes from Bourgain and Demeter's The proof of the $l^2$ Decoupling Conjecture. A related question was asked about a year ago go on MO, but the author was interested in the reverse ...

**0**

votes

**0**answers

55 views

### Is there an analogue of Fourier series using infinite series of Jacobi elliptic functions to model doubly-periodic functions?

We can use Fourier series to model general singly-periodic functions. Infinite sums of orthogonal trigonometric polynomials can approximate any singly-periodic function that is continuous along its ...

**0**

votes

**0**answers

48 views

### Optimality of Exponent in $\ell^{2}$ Decoupling Theorem

For $\delta>0$, let $\mathcal{N}_{\delta}$ denote the $\delta$-neighborhood of the truncated paraboloid $P^{n-1}$ and let $\theta$ denote a $\delta^{1/2}\times\cdots\times\delta^{1/2}\times\delta$ ...

**1**

vote

**0**answers

48 views

### Sobolev norm of a composition with a singular homeo

Let $H_p^t(\mathbb{R})$ be a fractional Sobolev space with the standard norm. The with $p>1$, $0<t<1$. Take some smooth $\phi$ from this space. Let $T$ be an ivertible homeomorphism of $\...

**3**

votes

**0**answers

86 views

### Optimal Kakeya Maximal Bound for Bushes

Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In ...

**2**

votes

**0**answers

131 views

### system of complex number equations

Let $a_1,a_2,a_3,a_4\in \mathbb{C}$ be distinct such that
$$a_1^3+a_2^3+a_3^3+a_4^3=0$$
$$(1+|a_1|^2)a_1^2+(1+|a_2|^2)a_2^2+(1+|a_3|^2)a_3^2+(1+|a_4|^2)a_4^2=0$$
$$(1+2|a_1|^2+2|a_1|^4)a_1+(1+2|a_2|^...

**1**

vote

**0**answers

42 views

### A question about Fourier transform of function of the type $(1+P(x))^{z}$

Let $$f= (1+P(x))^{z},$$
where $P(x)\ge 0$ is a real polynomial in $\mathbb{R}^n$ of degree $2m>0$, and $z=a+ib$ with $a<0$. I want to study the behavior near the origin of the Fourier ...

**0**

votes

**0**answers

82 views

### An integral form of sum $\sum_{n\geq 0} \frac{f^{(n)}(0) g^{(n)}(0)}{(n!)^2}$ for two real analytic at 0 functions? Fourier|Taylor series parallels

Thinking of parallels between Fourier series and Taylor series, you might find out that the integral $\frac 1 {2 \pi}\int\limits_0^{2 \pi} f(e^{it})\,\overline{g(e^{it})} \,dt=\langle f,\, g\rangle$ ...

**0**

votes

**0**answers

55 views

### Fourier analytic estimate

The following question arises naturally from applications to the image processing. Let $\alpha\in [0,1]$ and assume that for infinitely many $n\ge 1$ we have
$$\sum_{k=1}^n\frac{1-|\cos(2\pi k\alpha)|...

**2**

votes

**1**answer

152 views

### Character group of the multiplicative rationals

I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it here on math stack exchange first.
Let $\mathbb{Q}^{\times}$ be the multiplicative ...

**1**

vote

**1**answer

250 views

### Is this function $\mathbb{Q}$ periodic?

Considering a function $f$ exponentially decreasing at infinity, is the following function $\mathbb{Q}$ periodic ?
$$F(x)= \sum\limits_{q =1}^{\infty} \; \sum\limits_{n =1}^{\infty} \; \sum\...

**3**

votes

**1**answer

177 views

### On construction of a $\mathbb{Q}$ periodic function with Fourier series

Taking $f$ a function decreasing exponentially at infinity we can consider the periodic function given by following Fourier series:
$$F(x)= \sum\limits_{n =1}^{\infty} f(n) e^{2 i \pi n x}$$
Using ...

**1**

vote

**2**answers

68 views

### Solution to inhomogenous PDE

Given the equation $(1-\Delta)u=f$ for $f \in S(\mathbb{R}^n)$ (rapidly decreasing functions) we get by taking the Fourier transform that
$u = \left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\mathcal{F}^{-...

**5**

votes

**1**answer

211 views

### Fourier transform surjective on $L^p(\mathbb{R}^n)$ for $p \in (1,2)$?

I know that $F_2:L^2 \rightarrow L^2$ is of course unitary, whereas $F_1:L^1 \rightarrow C_0$ is injective but not surjective. This can be seen by looking at the dual map.
Riesz-Thorin gives us that ...

**11**

votes

**3**answers

1k views

### Eigenvectors of the Fourier transformation

The Fourier transform $\hat u$ is defined on the Schwartz space $\mathscr S(\mathbb R^n)$
by
$
\hat u(\xi)=\int e^{-2iπ x\cdot \xi} u(x) dx.
$
It is an isomorphism of $\mathscr S(\mathbb R^n)$ and the ...

**1**

vote

**1**answer

77 views

### Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...

**1**

vote

**0**answers

71 views

### Hilbert Transform and multiplier in $\mathbb{C}(X)$

I found myself trying to solve an equation of that kind :
$$
H f= R f,
$$
where $f$ has to be found in $L^2(\mathbb{R})$, $H$ is the Hilbert transform and $R$ is a rational function having no poles ...

**8**

votes

**1**answer

282 views

### Have the explicit Poisson-type formulas of Guinand and Meyer been observed before?

In a recent paper of Meyer Measures with locally finite support and spectrum PNAS vol. 113 no. 12:3152–3158 (behind a paywall, but see also these seminar notes) some new explicit Poisson-type formulas ...

**3**

votes

**2**answers

238 views

### Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...

**3**

votes

**1**answer

157 views

### How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here.
Say we have a stochastic process described by a ...

**2**

votes

**1**answer

210 views

### Does Fourier Algebra of locally compact group separate compact sets of the group?

Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...

**7**

votes

**1**answer

98 views

### Fast Fourier Transforms for non-trigonometric bases

The fast Fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other bases, e.g. orthogonal polynomial bases ...

**14**

votes

**2**answers

520 views

### Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?

I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...

**0**

votes

**0**answers

62 views

### Functions $f \in S(\mathbb{R})$ with slow decay rate?

I came across this question by thinking about whether there are functions $f \in S(\mathbb{R})$ that satisfy for all(!) $a>0$ that $f e^{a|.|^a} \notin L^{\infty},$ as somebody on math....

**2**

votes

**0**answers

45 views

### Can we say translation/dilation of the $L^p-$multiplier is again a $L^{p}-$multiplier?

Suppose that $m:\mathbb R \to \mathbb C$ such that
$\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}$ (where $C$ is some constant, $f\in L^{p}$). (That is, $m$ is an $L^{p}-$ multiplier) ...

**1**

vote

**0**answers

68 views

### A complicated integral inequality

How can we bound this integral:
$${\displaystyle \int_{-1}^{1}2\left[\dfrac{1}{4}-\dfrac{1}{4\left(1-\xi^{2}\right)}\left(1-\dfrac{\xi^{2}}{2}\right)^{2}\right]\left(\hat{f}\left(\xi\right)\right)^{2}...

**2**

votes

**1**answer

143 views

### Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story
I want to prove Euler's reflection formula by showing that
\begin{equation*}
f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s)
\end{equation*}
is constant, where $s = \sigma + it$. It's easy to see ...

**5**

votes

**1**answer

620 views

### Is the following integral nonzero?

Recently I met an integral as follow:
$$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq i<j\leq9}\sin\frac{\theta_i-\theta_j}{2}\right)\left(\prod\limits_{i=1}^9(1+\cos(\theta_i-\theta_{i+...

**2**

votes

**0**answers

381 views

### What is the Fourier transform of this function?

Consider the function
$$
f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.
$$
It is known that $f(x_1,x_2)\in L^2(\...

**1**

vote

**1**answer

99 views

### Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...

**1**

vote

**1**answer

157 views

### Fourier transform of tanh [closed]

tanh is not absolutely integrable, so a direct fourier transform does not exist. But even for the signum function, which is not absolutely intrgrable, we can get the fourier transform by applying some ...

**50**

votes

**1**answer

2k views

### Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.

**7**

votes

**0**answers

105 views

### Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...

**9**

votes

**1**answer

261 views

### Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains
$$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$
The standard ...

**2**

votes

**1**answer

51 views

### How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...

**1**

vote

**1**answer

130 views

### harmonic balance method for a 2-mass 3-spring system [closed]

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...

**6**

votes

**1**answer

126 views

### Positivity of power of positive PSD matrices

Background: Let $M$ be an $n\times n$ matrix with nonnegative entries. It is immediate that for any integer $k$, $M^k$ has nonnegative entries.
Suppose now that, on top of having nonnegative entries, ...